# Euler problems/21 to 30

### From HaskellWiki

(→[http://projecteuler.net/index.php?section=problems&id=21 Problem 21]: a solution) |
(→[http://projecteuler.net/index.php?section=problems&id=26 Problem 26]: a solution) |
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Line 60: | Line 60: | ||

Solution: | Solution: | ||

<haskell> | <haskell> | ||

− | problem_26 = | + | problem_26 = fst $ maximumBy (\a b -> snd a `compare` snd b) |

+ | [(n,recurringCycle n) | n <- [1..999]] | ||

+ | where recurringCycle d = remainders d 10 [] | ||

+ | remainders d 0 rs = 0 | ||

+ | remainders d r rs = let r' = r `mod` d | ||

+ | in case findIndex (== r') rs of | ||

+ | Just i -> i + 1 | ||

+ | Nothing -> remainders d (10*r') (r':rs) | ||

</haskell> | </haskell> | ||

## Revision as of 05:39, 3 April 2007

## Contents |

## 1 Problem 21

Evaluate the sum of all amicable pairs under 10000.

Solution: This is a little slow because of the naive method used to compute the divisors.

problem_21 = sum [m+n | m <- [2..9999], let n = divisorsSum ! m, amicable m n] where amicable m n = m < n && n < 10000 && divisorsSum ! n == m divisorsSum = array (1,9999) [(i, sum (divisors i)) | i <- [1..9999]] divisors n = [j | j <- [1..n `div` 2], n `mod` j == 0]

## 2 Problem 22

What is the total of all the name scores in the file of first names?

Solution:

-- apply to a list of names problem_22 :: [String] -> Int problem_22 = sum . zipWith (*) [ 1 .. ] . map score where score = sum . map ( subtract 64 . ord )

## 3 Problem 23

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

Solution:

problem_23 = undefined

## 4 Problem 24

What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?

Solution:

perms [] = [[]] perms xs = do x <- xs map ( x: ) ( perms . delete x $ xs ) problem_24 = ( perms "0123456789" ) !! 999999

## 5 Problem 25

What is the first term in the Fibonacci sequence to contain 1000 digits?

Solution:

valid ( i, n ) = length ( show n ) == 1000 problem_25 = fst . head . filter valid . zip [ 1 .. ] $ fibs where fibs = 1 : 1 : 2 : zipWith (+) fibs ( tail fibs )

## 6 Problem 26

Find the value of d < 1000 for which 1/d contains the longest recurring cycle.

Solution:

problem_26 = fst $ maximumBy (\a b -> snd a `compare` snd b) [(n,recurringCycle n) | n <- [1..999]] where recurringCycle d = remainders d 10 [] remainders d 0 rs = 0 remainders d r rs = let r' = r `mod` d in case findIndex (== r') rs of Just i -> i + 1 Nothing -> remainders d (10*r') (r':rs)

## 7 Problem 27

Find a quadratic formula that produces the maximum number of primes for consecutive values of n.

Solution:

problem_27 = undefined

## 8 Problem 28

What is the sum of both diagonals in a 1001 by 1001 spiral?

Solution:

problem_28 = undefined

## 9 Problem 29

How many distinct terms are in the sequence generated by a^{b} for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?

Solution:

problem_29 = length . group . sort $ [a^b | a <- [2..100], b <- [2..100]]

## 10 Problem 30

Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.

Solution:

problem_30 = undefined